+0  
 
-1
1
1
avatar+211 

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

 May 24, 2024
 #1
avatar+129511 
+1

Note that the sum of the first n terms =   n * a1  +  d * (n)(n-1) / 2     where n > 1

 

So

S10  =  10a1 + d (10)(9) / 2  =   10a1 + 45d  =  0         (1)

S20 =   20a1 + d(20)(19) / 2  =   20a1 + 190d = 1/5     (2)

 

Multiply (1) by -2  and add to (2)  giving

 

100d = 1/5

d = 1/500

 

To find  a1

10a1 + 45 (1/500)  = 0

10a1 = -45/500

a1 = -45 / 5000  =  -9 / 1000

 

So

 

S70  =   70 (-9/1000) +  (1/500) (70)(69) / 2  =    21 / 5  =  4.2

 

 

cool cool cool

 May 24, 2024

6 Online Users

avatar
avatar
avatar
avatar
avatar